Editing Partition of unity (section)

==Variant definitions==
Sometimes a less restrictive definition is used: the sum of all the function values at a particular point is only required to be positive, rather than 1, for each point in the space.  However, given such a set of functions <math>\{ \psi_i \}_{i=1}^\infty</math> one can obtain a partition of unity in the strict sense by dividing by the sum; the partition becomes <math>\{ \sigma^{-1}\psi_i \}_{i=1}^\infty</math> where <math display="inline">\sigma(x) := \sum_{i=1}^\infty \psi_i(x)</math>, which is well defined since at each point only a finite number of terms are nonzero. Even further, some authors drop the requirement that the supports be locally finite, requiring only that <math display="inline">\sum_{i = 1}^\infty \psi_i(x) < \infty</math> for all <math>x</math>.<ref>{{Cite book| last=Strichartz| first= Robert S.|url=https://www.worldcat.org/oclc/54446554|title=A guide to distribution theory and Fourier transforms |date=2003|publisher=World Scientific Pub. Co|isbn=981-238-421-9|location=Singapore|oclc=54446554}}</ref>

In the field of [[Operator algebra|operator algebras]], a partition of unity is composed of projections<ref>{{cite book |last1=Conway |first1=John B. |title=A Course in Functional Analysis |publisher=Springer |isbn=0-387-97245-5 |page=54 |edition=2nd}}</ref> <math>p_i=p_i^*=p_i^2</math>. In the case of [[C*-algebra|<math>\mathrm{C}^*</math>-algebras]], it can be shown that the entries are pairwise [[Orthogonality|orthogonal]]:<ref>{{cite book |last1=Freslon |first1=Amaury |title=Compact matrix quantum groups and their combinatorics |date=2023 |publisher=Cambridge University Press|bibcode=2023cmqg.book.....F }}</ref>
<math display="block">p_ip_j=\delta_{i,j}p_i\qquad (p_i,\,p_j\in R).</math>
Note it is ''not'' the case that in a general [[*-algebra]] that the entries of a partition of unity are pairwise orthogonal.<ref>{{cite web |last1=Fritz |first1=Tobias |title=Pairwise orthogonality for partitions of unity in a *-algebra|url=https://mathoverflow.net/a/463103/35482 |website=Mathoverflow |access-date=7 February 2024}}</ref>
 
If <math>a</math> is a [[normal element|normal]] element of a unital <math>\mathrm{C}^*</math>-algebra <math>A</math>, and has finite [[Spectrum (functional analysis)|spectrum]] <math>\sigma(a)=\{\lambda_1,\dots,\lambda_N\}</math>, then the projections in the [[Spectral theorem|spectral decomposition]]:
<math display="block">a=\sum_{i=1}^N\lambda_i\,P_i,</math>
form a partition of unity.<ref>{{cite book |last1=Murphy |first1=Gerard J. |title=C*-Algebras and Operator Theory |date=1990 |publisher=Academic Press |isbn=0-12-511360-9 |page=66}}</ref>

In the field of [[Compact quantum group|compact quantum groups]], the rows and columns of the fundamental representation <math>u\in
M_N(C)</math> of a quantum permutation group <math>(C,u)</math> form partitions of unity.<ref>{{cite book |last1=Banica |first1=Teo |title=Introduction to Quantum Groups |date=2023 |publisher=Springer |isbn=978-3-031-23816-1}}</ref>